Composite Functions: Monotonicity & Extrema

Help students master composite functions, monotonicity, and local extrema through structured calculus practice!
This NO PREP Composite Functions Worksheet Pack gives students focused practice analyzing increasing and decreasing intervals, applying the Chain Rule, interpreting derivative sign charts and graphs, and determining local extrema.
Students progress from foundational sign analysis to more challenging composite functions involving quadratic expressions, transformations, derivative graphs, nested functions, and absolute value.
What’s Included
15 pages of structured practice

• Composite functions and the Chain Rule
• Increasing & decreasing intervals
• Sign charts of (f’(x))
• Graph interpretation and derivative analysis
• Quadratic inner functions and transformations
• Finding and classifying local extrema
• Multi-step composite function problems
• Advanced challenge problems
• Detailed Answer Key included
• NO PREP — print and teach!
  Students will learn to:
  -Apply the Chain Rule to (g(x)=f(u(x)))
  -Analyze the signs of (f’(u(x))) and (u’(x))
  -Determine where composite functions increase or decrease
  -Interpret graphs and derivative sign charts
  -Identify critical numbers and local extrema
  -Analyze functions involving transformations and absolute value
  Solve increasingly complex calculus problems using multiple representations
  The worksheet progresses from guided skill-building to independent analysis and final challenge problems, making it useful for both instruction and practice.
  Perfect For
  -Precalculus
  -Calculus
  -AP Calculus AB Support
  -Grades 11–12
  -Advanced High School Math
  Use it for classwork, homework, review, independent practice, math centers, intervention, enrichment, or test preparation.
  Why Teachers Will Love It
  -Clear progression, varied problem types, student-friendly layouts, and built-in workspace make this resource easy to use in class. The included answer key also makes checking and review simple.
  -Just print and go — no additional prep required!

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